Computational Astronomy: Five Centuries of Finding True Syzygy

See, for example, Goldstein B. R. , “Descriptions of astronomical instruments in Hebrew”, in From deferent to equant: A volume of studies in the history of science in the ancient and medieval Near East in honor of E. S. Kennedy, ed. by King D. A. and Saliba G. (New York Academy of Sciences, New York, 1987), 105–41 (espec. p. 128, where al-Hadib's view is cited).

In addition to the authors whose works are discussed here, we know of a few other treatments of this question. The most important is probably that of John of Murs noted by Poulle E. , “John of Murs”, in Dictionary of scientific biography (New York, 1970–80), vii, 128–33 (espec. p. 130), and we have also located tables for this purpose in Hebrew astronomical manuscripts: Moses Farissol Botarel (c. 1481) in Munich, MS heb. 343, ff. 96b–97a, and Isaac ben Elia Kohen of Syracuse (c. 1491) in London, MS British Library Or. 2806, ff. 43a–44b.

Jensen C. , “The lunar theories of al-Baghdadi”, Archive for history of exact sciences, viii (1972), 321–8; King D. A. , “A double argument table for the lunar equation in Ibn Yūnus”, Centaurus, xviii (1974), 129–46; King D. A. , “Some early Islamic tables for determining lunar crescent visibility”, in King and Saliba (eds), op. cit. (ref. 1), 185–225; North J. D. , “The Alfonsine Tables in England”, in Prismata, ed. by Maeyama Y. and Saltzer W. G. (Wiesbaden, 1977), 269–301; Saliba G. , “The double argument lunar tables of Cyriacus”, Journal for the history of astronomy, vii (1976), 41–46; Tichenor M. , “Late medieval two-argument tables for planetary longitudes”, Journal of Near Eastern studies, xxvi (1967), 126–8, reprinted in Kennedy E. S. , Studies in the Islamic exact sciences (Beirut, 1983), 122–24.

See, for example, Goldstein B. R. , “A medieval table for reckoning time from solar altitude”, Scripta mathematica, xxvii (1964), 61–66, reprinted in Kennedy , Studies (ref. 3), 293–8.

Neugebauer O. , A history of ancient mathematical astronomy (New York and Berlin, 1975), 123, points out that Ptolemy does not indicate whether iteration is necessary, whereas in works of the Byzantine period it is explicitly stated that the procedure is to be iterated until no elongation remains. See also Nallino C. A. , Al-Battānī sive Albatenii opus astronomicum (2 vols, Milan, 1903–7), i, 94.

Neugebauer O. , The astronomical tables of al-Khwārizmī (Copenhagen, 1962), 63.

On Zijes (sets of astronomical tables), see Kennedy E. S. , A survey of Islamic astronomical tables, Transactions of the American Philosophical Society (Philadelphia, 1956), xlvi/2; see also Pedersen F. S. , “Canones Azarchelis: Some versions, and a text”, Cahiers de l'Institut du Moyen-Age grec et latin, liv (1987), 129–218 (espec. p. 182); Rico y Sinobas M. , Libros del saber de astronomía del Rey Alfonso X de Castilla (5 vols, Madrid, 1866), iv, 150–1.

Chabás J. and Goldstein B. R. , “Andalusian astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād”, Archive for history of exact science, xlviii (1994), 1–41.

Millás J. M. , Las tablas astronómicas del Rey Don Pedro el Ceremonioso (Madrid and Barcelona, 1962); Chabás J. , “Astronomía Andalusí en Cataluña: Las Tablas de Barcelona”, in From Baghdad to Barcelona: Studies in the Islamic exact sciences in honour of Prof. Juan Vernet, ed. by Casulleras J. and Samsó J. (Barcelona, 1996), 477–525; Chabás J. , L'astronomia de Jacob ben David Bonjorn (Barcelona, 1992), 23.

Poulle E. , Les Tables Alphonsines avec les canons de Jean de Saxe (Paris, 1984), 80ff.

Chabás J. and Goldstein B. R. , “Nicholaus de Heybech and his table for finding true syzygy”, Historia mathematica, xix (1992), 265–89.

Goldstein B. R. , The astronomical tables of Levi ben Gerson (New Haven, 1974), 136–46. For examples of medieval planetary correction tables where negative terms are eliminated by adding a constant, see Salam H. and Kennedy E. S. , “Solar and lunar tables in early Islamic astronomy”, Journal of the American Oriental Society, lxxxvii (1968), 492–7; Kennedy E. S. , “The astronomical tables of Ibn al-A^clam”, Journal for the history of Arabic science, i (1977), 13–23 (espec. p. 14); and Goldstein B. R. , “The survival of Arabic astronomy in Hebrew”, Journal for the history of Arabic science, iii (1979), 31–39 (espec. p. 37).

Goldstein B. R. , “Scientific traditions in late medieval Jewish communities”, in Les Juifs au regard de l'histoire: Mélanges en l'honneur de M. Bernhard Blumenkranz., ed. by Dahan G. (Paris, 1985), 235–47.

On al-Raqqām Ibn , see Carandell J. , Risāla fī 'ilm al-ẓilāl de Muhammad Ibn al-Raqqām al-Andalusī (Barcelona, 1988).

On Zacut , see Burgos Cantera F. , “El judío salmantino Abraham Zacut”, Revista de la Academia de Ciencias de Madrid, xxvii (1931), 63–398; and Burgos Cantera F. , Abraham Zacut (Madrid, 1935).

Heybech's entire table is published in Chabás and Goldstein , op. cit. (ref. 11), together with an explanation of the way it was computed.

For the use of Ptolemy's second lunar model in computing lunar velocities, see Goldstein B. R. , “Lunar velocity in the Ptolemaic tradition”, in The investigation of difficult things: Essays on Newton and the history of the exact sciences, ed. by Harman P. M. and Shapiro A. E. (Cambridge, 1992), 3–17.

Solon P. C. , “The ‘Hexapterygon’ of Michael Chrysokokkes”, Ph.D. dissertation, Brown University, 1968; Solon P. C. , “The six wings of Immanuel Bonfils and Michael Chrysokokkes”, Centaurus, xv (1970), 1–20.

This device is reminiscent of a similar one used by ben Gerson Levi for the same purpose relating to the same problem. Bonfils was aware of the work of his predecessor in southern France, and probably depended on him here: cf. Goldstein B. R. , The astronomy of Levi ben Gerson (1288–1344) (New York and Berlin, 1985), 9; see also ref. 12, above.

Nallino , op. cit. (ref. 5), ii, 80, 88.

Swerdlow N. M. and Neugebauer O. , Mathematical astronomy in Copernicus's De revolutionibus (New York and Berlin, 1984), 276.

Swerdlow and Neugebauer , op. cit. (ref. 21), 272.

Czartoryski P. , “The library of Copernicus”, Studia Copernicana, xvi (1978), 355–96 (espec. p. 366).

Tychonis Brahe Opera omnia, ed. by Dreyer J. L. E. (15 vols, Copenhagen, 1913–29), xii, 20–25.

Thoren V. E. , “Tycho Brahe's discovery of the variation”, Centaurus, xii (1967), 151–66 (espec. p. 158). Thoren also claimed that this observation led Brahe to the discovery of the lunar variation, but we do not believe there is enough evidence to support his view.

Poulle , op. cit. (ref. 10), 214ff; Chabás and Goldstein , op. cit. (ref. 11), 271.